


How Many Angels Can Fit On The Length Of A Pin?

by primeideal



Category: Gödel Escher Bach: An Eternal Golden Braid - Douglas Hofstadter
Genre: Gen, Mathematics, Measure Theory, script dialogue
Language: English
Status: Completed
Published: 2017-01-01
Updated: 2017-01-01
Packaged: 2018-09-13 20:37:48
Rating: General Audiences
Warnings: No Archive Warnings Apply
Chapters: 1
Words: 1,152
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/9141322
Author URL: https://archiveofourown.org/users/primeideal/pseuds/primeideal
Summary: Achilles, the Tortoise, and the Genie debate a not-exactly age-old question.





	

_Achilles and the Tortoise are passing the day with their friend the Genie. Or is this just a Genie isomorphic to the one they met several layers into another story altogether? He doesn’t seem to be particular about the distinction._

Achilles: It’s been a pleasure, but we should be going. I think it’s my friend the Sloth’s birthday soon, so I need to buy him a present.

Tortoise: You _think_ it’s his birthday? Because we can try and investigate!

Achilles: Oh, don’t you start.

Genie: I’ll leave you to it. My fellow djinn and I have been having some trouble with some angels, and I ought to be cleaning that up, if I can.

Tortoise: What sort of trouble?

Genie: You see this pin?

_He carefully indicates a pin that has been lingering near his lamp. Like the lamp, it’s golden and shimmers briefly in the light._

Achilles: Obviously.

Genie: Some angels have been crowding onto it. Rubbing shoulder-to-shoulder, mind you, so they all fit in. But we can’t figure out where they come from, what they want, or even how many of them there are?

Tortoise: Have you tried counting them all?

Genie: We _tried_ , yes.

Achilles: What happened?

Genie: I asked them to spread out. If they’d each take a place on one point of this continuum here, none overlapping and no space left empty, then the pin would be quite full of angels, and we could carry it somewhere more appropriate. But try as they might, no matter how they positioned themselves, they couldn’t fill it up.

Achilles: So there are fewer of them than there are points on the pin.

Tortoise: Yes. But why should that matter? We should be thinking of more important things. How much do they weigh? How far can they fly? What music do they like dancing to?

Achilles: I say, Genie? Do you still have your Meta-Lamp to summon the Meta-Genie?

Genie: Of course!

Achilles: And she in turn can summon the Meta-Meta-Genie?

Genie: Quite so, quite so, and so on! We’re quite the clan of djinn, you know.

Achilles: Well, why don’t you all have a little reunion, and then each of you can discuss matters with your own angel?

Genie: Well, there might be more of them than we know what to do with.

Achilles: I thought they were fewer than the points on the pin.

Genie: Perhaps there’s some number in between.

Achilles: I can’t conceive of any such thing.

Genie: Well, maybe you’re wrong.

Achilles: Maybe _you’re_ wrong!

Genie: I say, don’t poke yourself!  


_For the Tortoise has cautiously picked up the pin by its end and is holding it at arm’s length._

Tortoise: It seems to me you ought to be concerned with how to feed and clothe all these angels.

Genie: What do you propose?

Tortoise: Clothing is simple. Certainly they’ll want to wear halos.

Genie: If I had a halo that was a pin’s length long, they’d all fit nicely inside.

Tortoise: With room to spare!

Genie: Since they don’t take up all the space.

Tortoise: Now, you and your fellow djinn are a different matter. If I had even the tiniest halo, why, I’d cut it in half, and wrap half of it around you. You’re a very skinny genie when you hover just like that, you know.

Genie: I’m flattered.

Tortoise: And I’d take what I had left, cut it in half again, and wrap _that_ around your friend Meta-Genie.

Genie: She’d look dashing.

Tortoise: And so on, and so forth. And no matter how small my halo was, I’d cut it and wrap it up for all of you!

Genie: That’s as may be, but what about the angels?

Tortoise: Well, what do healthy angels eat?

Genie: Angel food cake.

Tortoise: I should have known. Well, say the pin was full of nice, healthy angels. I’d wrap them all up in a halo a pin’s-length long. Or, I’d cut a nice slice of cake just that length, and it would go nicely inside the angels’ bellies.

Genie: But we djinn are skinny enough that we don’t eat much.

Tortoise: No. You’re healthy, with just as little room inside and out.

Achilles: Are there such things as unhealthy angels?

Tortoise: I’m afraid so. It’s positively demonic, really.

Genie: Then what of our friends on the pin?

Tortoise: Well, if there are only as many of them as there are you djinn—

Achilles: And that really seems the most sensible thing—

Tortoise: Then of course I could slice up the halos just as I was saying before.

Genie: And if there are more and more angels? Choirs full in patterns you can’t quite see yet, their unfamiliar harmonies echoing to the skies?

Tortoise: Well, either they’d be so sadly unhealthy I wouldn’t know where to begin, or...let’s see here…

_He turns the pin over, squinting in the glow._

Tortoise: If they’re healthy enough, they’d _still_ be just as small and skinny as you all, and then they might as well crowd onto the head of the pin! I don’t see what all the fuss is about.

Achilles: I don’t care what sort of clothes they wear, I care whether the djinn can pair off with them. You’re just scared they’ll try to set you up on dates if you correspond nicely.

Genie: Poppycock! You’re just unable to admit there are all sorts of choirs you can’t compare to anything familiar.

Achilles: I’m talking to a djinn while my friend the tortoise holds a pin with a choir of angels on it. If there were any reason to believe these sorts of choirs were out there, I think I could come to grips with the news.

Tortoise: Have the other djinn tried to figure this out?

Genie: Meta-Genie has worked rather hard on it, but to no avail.

Achilles: I’m sure she’ll get there.

Tortoise: And if she doesn’t? Maybe instead of trying to imagine these choirs of angels, you should try to imagine two different worlds; one where they can pair off with the djinn, one where they can’t.

Achilles: Two worlds? Surely they can’t both be accessible?

Genie: They might as well be equally accessible.

Tortoise: This reminds me of a series of woodcuts by Escher I’ve been meaning to revisit. The _Circle L_ _i_ _mits_ ; do you know them?

Achilles: Oh, yes! The fourth is my favorite.

Tortoise: Technically, I think they should be called _Hyperbolic Limits,_ but—

Achilles: Hyperbolic space! Such a heavenly space to visit.

Tortoise: It’s so cozy!

Achilles: Indeed. Through any line and a point not on it, there exists no line through the point parallel to the given line. Lovely place to spend an afternoon.

_A passing mockingbird swoops down._

Mockingbird: I think you meant, through any sentence—

Genie: Buzz off, Queneau, no one cares. And give me my pin back before you poke your eye out.

**Author's Note:**

> The first draft of this was written in 2013 with a few edits made since then. Huge thanks to L. for the encouragement and support needed to get this written, S. for measure theory knowledge, and to the good people of Yuletide for reminding me that yes, there's a fandom for everything.


End file.
